Linking integrals in the n-sphere

TL;DR

This paper introduces a new linking integral for disjoint submanifolds in the n-sphere by relating their join to the linking number, providing a novel geometric approach.

Contribution

It defines a map from the join of submanifolds to the n-sphere that captures the linking number as its degree, offering a new method to compute linking integrals.

Findings

Established a map from the join of submanifolds to the n-sphere

Linked the degree of this map to the linking number

Provided a new integral formula for linking in the n-sphere

Abstract

Let K and L be disjoint closed oriented submanifolds of the n-sphere, with dimensions adding up to n-1. We define a map from their join K*L to the n-sphere whose degree up to sign equals their linking number, and then use this to find the desired linking integral.